4.4 Variational Problems of Functionals with Functions of Several Variables
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obtains extremum, the additional condition is that the boundaries of the surfaces
u(x, y), v(x, y), w(x, y) are on a given surface, namely
Φ(u, v, w) = 0
(4.4.2)
where, Φ has enough partial derivative, then there is the boundary condition
F u x F u y Φ u
F v x F v y Φ v
F w x F w y Φ w
= 0
(4.4.3)
The determinant (4.4.3) is called the Courant condition. The courant condition
can be popularized to higher-order determinant. Generally speaking, for the functional with the n-tuple integral and first partial derivative, the corresponding Courant
condition is that the n + 1 order determinant is zero.
Proof Let on coordinate plane Ox y, the boundary line of the domain D be a curve
Γ , and let (x, y) be the same as the rotating direction of the coordinate axis of (u, v).
Obviously, when Φ(u, v, w) = 0, the integral is
Γ
Φ(u(x, y), v(x, y), w(x, y))dΓ = 0
(4.4.4)
Thus, the extremal problem of the functional (4.4.1) under the additional condition
(4.4.2) is converted into finding the necessary condition of the extremum of the
functional (4.4.1) under the additional condition (4.4.4).
Using Lagrange multiplier method and introducing the parameter ε = (ε 1 , ε 2 , ε 3 ),
such that δu = ε 1 ξ , δv = ε 2 η, δw = ε 3 ζ , …, δw y = ε 3 ζ y , the exreamal problem of
the functional (4.4.1) is converted into the exreamal problem of the functional
J
∗
(ε, λ) =
¨
D
F(u + ε 1 ξ, v + ε 2 η, w + ε 3 ζ, u x + ε 1 ξ x , . . . , w y + ε 3 ζ y )dxdy
+ λ
Γ
Φ(u(x, y), v(x, y), w(x, y))dΓ
(4.4.5)
at ε = 0. Where, λ is called the Lagrange multiplier, ξ , η, ζ are arbitrary second
order differentiable functions of x and y.
By the necessary condition of the extremum of the functional, the following
equations can be obtained